Five-dimensional gauge theories and local mirror symmetry

We study the dynamics of 5-dimensional gauge theory on M-4 x S-1 by compact ifying type II/M theory on degenerate Calabi-Yau manifolds. We use the loca l mirror symmetry and shall show that the prepotential of the 5-dimensional SU(2) gauge theory without matter is given exactly by that of the type II string theory compactified on the local F-2, i.e., Hirzebruch surface F-2 l ying inside a noncompact Calabi-Yau manifold. It is shown that our result r eproduces the Seiberg-Witten theory at the 4-dimensional limit R --> 0 (R d enotes the radius of S-1) and also the result of the uncompactified 5-dimen sional theory at R --> infinity. We also discuss SU(2) gauge theory with 1 less than or equal to N-f less th an or equal to 4 matter in vector representations and show that they are de scribed by the geometry of the local F-2 blown up at N-f points. (C) 2000 E lsevier Science B,V. All rights reserved.